Uploaded March 2023 | Updated September 2026, 1 week ago
Previous tensor product video: youtu.be/KnSZBjnd_74
The universal property of the tensor product is one of the most important tools for handling tensor products. It gives us a way to define functions on the tensor product using bilinear maps. However, the statement of the universal property can be confusing if it is presented without background. This video is an explanation of the universal property that proves it for a concrete instantiation of the tensor product of vector spaces (or modules over a commutative ring).
Tensor Products playlist: youtube.com/playlist?list=PLug5ZIRrShJHCtzgzZyRqSdzr8wYlN2qk
0:00 Introduction
3:04 Constructing the Tensor Product
7:54 Bilinear Maps
10:39 Maps on the Tensor Product
16:17 Defining g
26:24 Linearity and Uniqueness
29:44 Universal Property
30:50 Example
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Music: C418 - Pr Department
Previous tensor product video: youtu.be/KnSZBjnd_74
The universal property of the tensor product is one of the most important tools for handling tensor products. It gives us a way to define functions on the tensor product using bilinear maps. However, the statement of the universal property can be confusing if it is presented without background. This video is an explanation of the universal property that proves it for a concrete instantiation of the tensor product of vector spaces (or modules over a commutative ring).
Tensor Products playlist: youtube.com/playlist?list=PLug5ZIRrShJHCtzgzZyRqSdzr8wYlN2qk
0:00 Introduction
3:04 Constructing the Tensor Product
7:54 Bilinear Maps
10:39 Maps on the Tensor Product
16:17 Defining g
26:24 Linearity and Uniqueness
29:44 Universal Property
30:50 Example
Subscribe to see more new math videos!
Music: C418 - Pr Department
![Matrix Invertibility With F[x]-Modules](https://i.ytimg.com/vi/xnZ2d6y8JmA/mqdefault.jpg "Matrix Invertibility With F[x]-Modules
Characteristic polynomial explanation: https://youtu.be/jCt6mR3QtPk
Ring & Module Theory playlist: https://youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
In this video, we use F[x]-modules to prove that a matrix is invertible if and only if its determinant is nonzero. Another example of how F[x]-modules can be applied to a ton of things in linear algebra!
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Music: OcularNebula - The Lopez")
